Based on CB chat, there's some confusion with my solution, so I'll elaborate.

It's my understanding that you want to do something along the lines of

while (<STDIN> or <TNIN>) {
   if (we_read_from_STDIN()) {
      print TNOUT do_this($_);
   } else {
      print STDOUT do_that($_);
   }
}
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Of course, that's not valid Perl. That where IO::Select comes in.

use IO::Select qw( );

my $sel = IO::Select->new(\*STDIN, \*TNIN);

MAIN_LOOP:
while (my @ready = $sel->can_read()) {
   for my $fh (@ready) {
      my $bytes_read = sysread($fh, my $data='', 4096)
         or last MAIN_LOOP;

      if ($fh == \*STDIN) {
         print TNOUT do_this($data);
      } else {
         print STDOUT do_that($data);
      }
   }
}
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We can't use <$fh> since that would block when only part of a line has arrived. If you want to do line-based IO, you'll have to do your own line building:

use IO::Select qw( );

my $sel = IO::Select->new(\*STDIN, \*TNIN);
my %bufs;

MAIN_LOOP:
while (my @ready = $sel->can_read()) {
   for my $fh (@ready) {
      our $buf; local *buf = \$bufs{$fh};  # Alias

      my $bytes_read = sysread($fh, $buf, 4096, length($buf))
         or last MAIN_LOOP;

      my $line_end = index($buf, $/);
      next if $line_end < 0;

      my $line = substr($buf, 0, $line_end+1, '');

      if ($fh == \*STDIN) {
         print TNOUT do_this($line);
      } else {
         print STDOUT do_that($line);
      }
   }
}
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Update:

• I'm not sure about the $fh == \*STDIN part. fileno($fh) == fileno(\*STDIN) should definitely work.
• Note: %bufs may still contain data when last MAIN_LOOP; is executed.
• Bonus: Since can_read accepts a timeout argument, you can probably do away with your alarm!
• Won't work on Windows.